Diagonal and Monomial Solutions of the Matrix Equation A X B = C

In this article, we consider the matrix equation AXB = C, where A, B, C are given matrices and give new necessary and sufficient conditions for the existence of the diagonal solutions and monomial solu- tions to this equation. We also present a general form of such solutions. Moreover, we consider the least squares problem minX kC ? AXBkF where X is a diagonal or monomial matrix. The explicit expressions of the optimal solution and the minimum norm solution are both provided.